Adiabatic regularization with a Yukawa interaction

Abstract

We extend the adiabatic regularization method for an expanding universe to include the Yukawa interaction between quantized Dirac fermions and a homogeneous background scalar field. We give explicit expressions for the renormalized expectation values of the stressenergy tensor T µν and the bilinear ψ ψ in a spatially flat Friedmann-Lemaitre-RobertsonWalker (FLRW) spacetime. These are basic ingredients in the semiclassical field equations of fermionic matter in curved spacetime interacting with a background scalar field. The ultraviolet subtracting terms of the adiabatic regularization can be naturally interpreted as coming from appropriate counterterms of the background fields. We fix the required covariant counterterms. To test our approach we determine the contribution of the Yukawa interaction to the conformal anomaly in the massless limit and show its consistency with the heat-kernel method using the effective action.

Adiabatic expansions for Dirac fields, renormalization, and anomalies

“…We have systematically scrutinized it and shown that its predictions for the (renormalization) subtraction terms match exactly those of the approach discussed in Refs. [14,15,17,18]. In particular we have computed the conformal and axial anomalies emerging for massless fermions and found an exact agreement with the results obtained by other methods.…”

“…If the classical field Φ is also assumed to be homogeneous Φ ¼ ΦðtÞ, as the classical background geometry, the modes for the quantized Dirac field can be expressed as above. However, the basic equations (3.6) are now replaced by [17] i∂ where sðtÞ ≔ g Y ΦðtÞ.…”

“…In order to work out the conformal anomaly (see [17] and later on), we need to compute hðtj4Þ by using (4.12). It is important to keep in mind that the relevant part of U 4 ðt; t 0 Þhðt 0 Þ is just the phase exp ð−i R t ω 4 Þ which drops out of the computations of the anomaly, which is closely related to hψψi.…”

“…In particular, in order to understand the reheating process after inflation, Ref. [17] analyzes the Yukawa coupling of a quantized Dirac field to an external scalar. Also, in Ref.…”

Chiral gravitational waves from chiral fermions

“…However, we warn the reader that these equations are contaminated by ultraviolet (UV) divergences that have to be exorcised before making sense of their physical significance. Unlike the bosonic UV divergences, UV divergences of fermions in FRW background is a subtle topic that has been discussed in a few works, see, e.g., [50][51][52], and in the context of GW production in [46,49]. In this work we do not try to follow a rigorous procedure to regularize expressions (32) and (33).…”